Efficient Sampling on Lattices

We will survey some old, some new, and some open problems in the area of efficient sampling. We will focus on sampling combinatorial structures (such as perfect matchings and independent sets) on regular lattices. These problems arise in statistical physics, where sampling objects on lattices can be used to determine many thermodynamic properties of simple physical systems. For example, perfect matchings on the Cartesian lattice, more commonly referred to as domino tilings of the chessboard, correspond to systems of diatomic molecules. But most importantly they are just cool problems with some beautiful solutions and a surprising number of unsolved challenges!